Wigner functions defined with Laplace transform kernels

Se Baek Oh; Jonathan C. Petruccelli; Lei Tian; George Barbastathis

doi:10.1364/OE.19.021938

1. Introduction

Since the Wigner function (WF) was formulated as a quasi–probability distribution function in quantum mechanics [1], it has been extremely useful in various optical applications such as first–order optical systems modeling [2], partially coherent beams propagation [3], and spatio–temporal beam characterization [4]. In the signal–processing community, the WF is known as one of the time–frequency distribution functions [5] or the simplest form of Cohen’s class [6], and the WF was used for signal analysis and synthesis [7].

For a signal ψ(q), typically a scalar field in optics, the WF is defined as

𝒲 (q, p) = \int_{- \infty}^{\infty} ψ (q + \frac{q^{'}}{2}) ψ^{*} (q - \frac{q^{'}}{2}) e^{- i q^{'} \cdot p} d q^{'},

where q is a real primary variable – typically space or time – and p is the corresponding real momentum variable – spatial frequency or temporal frequency. The WF describes signals or systems in p and q simultaneously – often called phase–space. Among many properties of the WF, its marginals are particularly useful: (1) the double integration of the WF over both p and q is proportional to the energy of the original signal ψ(q), (2) the integration along p is proportional to |ψ(q)|² – the intensity of ψ(q), and (3) the integration along q is proportional to the spatial power spectral density |Ψ(p)|², where Ψ(p) is the Fourier transform of ψ(q). Because the WF exhibits many useful properties and represents signals in phase–space, the WF provides new aspects and intuitions on optical signals and systems [8].

Note that Eq. (1) is indeed the Fourier transform of a correlation function, which is Hermitian; hence, the WF is always real. Let us consider a more general form of Eq. (1) based on the fact that the Fourier transform is a special case of the Laplace transform. Specifically, the kernel in the Laplace transform can be written as e^−s·q′, where s = p_R + ip_I, and if p_R = 0, then the Laplace and the Fourier kernels become identical. The Laplace transform supports a broader type of functions, even for a certain function whose Fourier transform does not exist, such as a exponentially increasing function. Replacing the Fourier kernel in Eq. (1) with the Laplace kernel allows us to analyze various complex signals in phase–space and provides a more general description of signals.

In this paper, we generalize the WF to a Laplace kernel Wigner Function (LWF) by allowing momentum variables to take on complex values. One potential application is analyzing evanescent waves or inhomogenous waves, whose envelopes vary exponentially, as we will show later.

2. Laplace kernel Wigner Function

For simplicity, we consider a scalar field ψ described in one–dimensional spatial coordinate x. We define the Laplace kernel Wigner Function as

𝒲_{ℒ} (x, s) = \int_{- \infty}^{\infty} ψ (x + \frac{x^{'}}{2}) ψ^{*} (x - \frac{x^{'}}{2}) e^{- s x^{'}} d x^{'},

where s = p_R + ip_I. As the range of the integration is from −∞ to ∞, Eq. (2) is the bilateral Laplace transform [9]. If we treat p_R and p_I independently, the LWF is illustrated in 3D space (x–p_R–p_I) as shown in Fig. 1. One distinction between the Laplace and Fourier transforms is the Region Of Convergence (ROC) S, which is the region of Re{s} where the Laplace transform exists. Hence, the LWF is confined inside the ROC along the p_R axis, as shown in Fig. 1.

Fig. 1 The LWF is visualized in x–p_R–p_I space. Due to the ROC, the LWF may be limited in p_R. The extents along x and p_I are determined by the space–bandwidth product of signals. At p_R = 0, the x–p_I plane in the LWF corresponds to the x–p space in the traditional WF.

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Taking the inverse Laplace transform, often called the Bromwich integral or Fourier–Mellin integral [10], allows us to get the original correlation function from the LWF.

ψ (x_{1}) ψ^{*} (x_{2}) = \frac{1}{2 π i} \int_{γ - i \infty}^{γ + i \infty} 𝒲_{ℒ} (\frac{x_{1} + x_{2}}{2}, s) e^{s (x_{1} - x_{2})} d s,

where γ is a vertical contour in the complex plane chosen within the ROC. The derivation can be found in the supplement.

Note that if p_R = 0, then Eq. (2) becomes identical to the traditional WF as in Eq. (1). This implies that p_I at p_R = 0 corresponds to the momentum p of the traditional WF; hence, the x–p_I plane in the LWF is equivalent to the x–p plane in the traditional WF as shown in Fig. 1. Also all properties of the traditional WF hold in the x–p_I plane in the LWF.

We further explored the marginals of the LWF and summarized some of them in Table 2. Detailed proofs and more properties can be found in the supplement.

Table 1. Marginals of the WF and LWF. E is the energy of signals. C = ∫_𝒮 dp_R, which is a scaling factor dependent on the width of the ROC, if |𝒮| < ∞. ℒ_ψ is the Laplace transform of ψ. (†): if p_R ∈ 𝒮.

View Table

3. Example: LWF of evanescent waves in SPP

A surface plasmon polariton (SPP) is a collective electron plasma oscillation in a metal excited by phase–matched TM–electromagnetic waves [11]. As shown in Fig. 2, the SPP exhibits exponential decay in its transverse magnetic field component as

H_{y} (x, z) = {\begin{array}{l} A e^{i β x} e^{- k_{1} z} & if z \geq 0 \\ A e^{i β x} e^{k_{2} z} & if z < 0 \end{array},

where A is the amplitude of the evanescent waves at z = 0. The propagation constant β given by

β = k_{0} \sqrt{\frac{ɛ_{1} ɛ_{2}}{ɛ_{1} + ɛ_{2}}},

where ɛ₁ and ɛ₂ are the permittivity of air and metal, respectively [11], k₀ is ω/c, where ω is the angular frequency of the incident light and c is the speed of light. The k–vectors along z in air and metal are

k_{1}^{2} = β^{2} - k_{0}^{2} ɛ_{1} and

k_{2}^{2} = β^{2} - k_{0}^{2} ɛ_{2},

respectively. Since Eq. (4) is a separable function in x and z, we may peform the LWT along these axes independently. In general, the k_i and β are complex, since ɛ is complex-valued within the metal. However, for the examples considered here, the imaginary parts of k_i and β are small, and we neglect them for simplicity. The result of this approximation is that the LWF along x is identical to the traditional WDF because the ROC is Re{s} = 0. We therefore compute and plot the LWT along the z axis, which yields a three-dimensional distribution in p_r, p_i, and z. For the more realistic case of complex k_i and β, the LWT would be six-dimensional.

Fig. 2 Evanescent waves in SPP excited at the metal–air interface.

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By using Eq. (2), we obtain the LWF of the evanescent wave as

𝒲_{ℒ} (z, s) = e^{- 2 k_{i} | z |} {\frac{e^{2 s | z |}}{\frac{k_{1} + k_{2}}{2} - s} + \frac{e^{- 2 s | z |}}{\frac{k_{1} + k_{2}}{2} + s} + \frac{e^{2 s | z |} - e^{- 2 s | z |}}{s}},

where

k_{i} = {\begin{array}{l} k_{1} & if z \geq 0 \\ k_{2} & if z < 0 \end{array} .

The ROC is −(k₁ + k₂)/2 < p_R < (k₁ + k₂)/2. Detailed derivations can be found in the supplement. For visualization purposes, we illustrate the LWF of H_y(z) for k₁/k₀ = 0.336 and k₂/k₀ = 3.308 in Fig. 3. The values of k₁/k₀ and k₂/k₀ are chosen to represent a plasmon generated by a field whose vacuum wavelegth is 500 nm and the subscripts 1 and 2 denote quantities in air and silver, respectively.

Fig. 3 (a) [ Media 1] The real part and (b) [ Media 2] the complex part of the LWF for the SPP given by Eq. (8) for k₁/k₀ = 0.336 and k₂/k₀ = 3.308.

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We also compute the traditional WF for the same parameters and plot in Fig. 4. Note that the traditional WF appears at p_R = 0 in Fig. 3.

Fig. 4 Traditional WDF of evanescent waves for k₁/k₀ = 0.336 and k₂/k₀ = 3.308. Absolute values are plotted.

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4. Cross–terms in LWF

Due to the bilinear nature of the WF, the LWF also produces cross–terms (sometimes also called interference–terms) when two or more fields are added. Specifically, for two arbitrary functions f(z) and g(z),

\begin{array}{l} 𝒲_{ℒ} [f (z) + g (z)] = 𝒲_{ℒ} [f (z)] + 𝒲_{ℒ} [g (z)] \\ + \int_{- \infty}^{\infty} f (z + \frac{z^{'}}{2}) g^{*} (z - \frac{z^{'}}{2}) e^{- s z^{'}} d z^{'} + \int_{- \infty}^{\infty} g (z + \frac{z^{'}}{2}) f^{*} (z - \frac{z^{'}}{2}) e^{- s z^{'}} d z^{'} . \end{array}

We now compute one of the cross–terms in the specific case where

f (z) = {\begin{array}{l} e^{- k_{1} z}, & if z \geq 0 \\ e^{k_{2} z}, & if z < 0 \end{array}

and

g (z) = {\begin{array}{l} e^{- k_{3} z}, & if z \geq 0 \\ e^{k_{4} z}, & if z < 0 \end{array},

as in the evanescent waves of the SPP. The result is

\begin{array}{l} 𝒲_{ℒ f g} (z, s) = \int_{- \infty}^{\infty} f (z + \frac{z^{'}}{2}) g^{*} (z - \frac{z^{'}}{2}) e^{- s z^{'}} d z^{'} \\ = e^{- k_{m} | z |} \frac{e^{2 (k_{n} - s) | z |} - e^{- 2 (k_{n} - s) | z |}}{k_{n} - s} \\ + e^{(- k_{1} + k_{4}) z} \frac{e^{- 2 (\frac{k_{1} + k_{4}}{2} + s) | z |}}{\frac{k_{1} + k_{4}}{2} + s} + e^{(k_{2} - k_{3}) z} \frac{e^{- 2 (\frac{k_{2} + k_{3}}{2} - s) | z |}}{\frac{k_{2} + k_{3}}{2} - s}, \end{array}

where

k_{m} = {\begin{array}{l} k_{1} + k_{3} & if z \geq 0 \\ k_{2} + k_{4} & if z < 0 \end{array}

and

k_{n} = {\begin{array}{l} \frac{- k_{1} + k_{3}}{2} & if z \geq 0 \\ \frac{k_{2} - k_{4}}{2} & if z < 0 \end{array},

and the ROC is

- \frac{k_{1} + k_{4}}{2} < p_{R} < \frac{k_{2} + k_{3}}{2}

. The other cross–term can be computed easily by swapping f and g. When multiple evanescent waves interfere, these cross–terms would appear in the LWF. In Fig. 5, we plot the LWF of the cross–term for two evanescent waves, where k₁/k₀ = 0.336, k₂/k₀ = 3.308, k₃/k₀ = 0.225 and k₄/k₀ = 4.668. Here, k₁ and k₂ are chosen to match the previous example, while k₃ and k₄ represent a plasmon at the air-silver interface whose vacuum field has a wavelength of 650 nm, with the subscripts 3 and 4 representing air and silver, respectively.

Fig. 5 The LWF cross–terms of evanescent waves in SPP for k₁/k₀ = 0.336, k₂/k₀ = 3.308, k₃/k₀ = 0.225 and k₄/k₀ = 4.668. In (a) [ Media 3], the only one cross–term is shown whereas in (b) [ Media 4] the sum of the two cross–terms are shown. Real values are plotted.

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5. Conclusion

In this paper, we proposed the Laplace kernel Wigner Function, a new Wigner–type phase–space function with Laplace kernels. By the nature of the Laplace transform, the momentum variable in the LWF is allowed to be a complex number as well as the Region of Convergence should be accompanied. We explored some of the properties of the LWF, especially marginals and cross–terms. As an example, we analyzed the LWF of evanescent waves in SPP and visualized in new complex phase–space.

Although the LWF is more generalized than the WDF, it is not evident that the meaning of p_r, especially when it is not zero. Non–zero p_r provides information about the decay nature of signals because the Laplace transform essentially decomposes the correlation function into complex exponentials. The physical meaning of complex LWF is also not clear, although mathematically the LWF corresponds to the phase and amplitude of the decomposed complex exponentials. We can speculate that the LWF can be loosely interpreted as energy–like distribution between position and complex momentum space because the LWF satisfies appropriate marginals.

Our approach, allowing complex momentum variables, may be applicable for other higher–dimensional phase–space functions or time–frequency distribution functions as well. Investigating the behavior of partially coherent evanescent waves with the LWF is another interesting direction of future research.

Acknowledgments

Financial support was provided by Singapore’s National Research Foundation through the Centre for Environmental Sensing and Modeling (CENSAM) independent research groups of the Singapore- MIT Alliance for Research and Technology (SMART)

References and links

1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]

2. M. J. Bastiaans, “Wigner distribution function and its application to 1st-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]

3. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef]

4. J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995). [CrossRef]

5. B. Boashash, Time Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier Science Ltd, 2003).

6. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989). [CrossRef]

7. G. Boudreaux-Bartels, “Time-varying signal processing using Wigner distribution synthesis techniques,” in The Wigner Distribution–Theory and Applications in Signal Processing, W. Mecklenbruker and F. Hlawatsch, eds., (Elsevier Science, 1997), pp. 269–317. [PubMed]

8. M. J. Bastiaans, “Wigner distribution in optics,” in Phase-Space Optics: Fundamenals and Applications, M. Testorf, B. Hennelly, and J. Ojeda-Castañeda, eds. (McGraw-Hill, 2009), pp. 1–44.

9. A. Oppenheim, A. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

10. G. Arfken and H. Weber, Mathematical Methods for Physicists (Academic Press, 1995).

11. S. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).

WF 𝒲 (x, p)		LWF 𝒲_ℒ (x, s)
∫dp	\|ψ(x)\|²	∫dp_I	\|ψ(x)\|²	(†)
∫dp	\|ψ(x)\|²	∬dp_Rdp_I	C · \|ψ(x)\|²
∫dx	\|Ψ(p)\|²	∫dx	$ℒ_{ψ} (s) ℒ_{ψ}^{} (- s^{})$	(†)
∬dxdp	E	∬dxdp_I	E	(†)
∬dxdp	E	∭dxdp_Rdp_I	C · E

Wigner functions defined with Laplace transform kernels

Abstract

1. Introduction

2. Laplace kernel Wigner Function

3. Example: LWF of evanescent waves in SPP

4. Cross–terms in LWF

5. Conclusion

Acknowledgments

References and links

Supplementary Material (4)

Cited By

Figures (5)

Tables (1)

Equations (15)

Optics Express